Cost of Carry and Forward Pricing

Every lesson so far has taken the delivery price K as a given number written into the contract. Where does that number actually come from? Not from anyone's forecast of where the price is headed — from a pure no-arbitrage argument. A bullion bank quoting a six-month gold forward, an index desk quoting a forward on the S&P 500, an FX desk quoting a 90-day euro forward — all of them reach for the same formula, built from nothing more than the cost of borrowing money and holding the asset until delivery. That formula is the cost of carry, and it is the single most important result in this module.

The cash-and-carry arbitrage argument

Consider an investment asset — one held for investment rather than consumption, and freely available to buy, hold, or (for this argument) sell short — paying no income over the life of the contract: a non-dividend stock, or gold treated as a pure store of value. Let S_0 be today's spot price, r the continuously-compounded risk-free rate, and T the time to maturity in years. Suppose the market quotes a forward price F_0 for delivery at T. Two cases:

Case 1 — F_0 is too high, i.e. F_0 > S_0 e^{rT}. An arbitrageur borrows S_0 at rate r, buys the asset today, and simultaneously sells (goes short) a forward at F_0. At maturity they deliver the asset for F_0 and repay the loan, now grown to S_0 e^{rT}. Riskless profit, locked in today with zero net investment:

F_0 - S_0 e^{rT} \;>\; 0.

Every arbitrageur in the market does this at once — buying spot pushes S_0 up, selling forward pushes F_0 down, until the free profit disappears.

Case 2 — F_0 is too low, i.e. F_0 < S_0 e^{rT}. Run it in reverse — a reverse cash-and-carry: short-sell the asset for S_0, invest the proceeds at r, and go long a forward at F_0. At maturity the investment has grown to S_0 e^{rT}; use F_0 of it to take delivery under the forward and close out the short position, pocketing the rest:

S_0 e^{rT} - F_0 \;>\; 0.

Neither case can persist — so the only forward price that admits no riskless arbitrage is the one where both differences are exactly zero.

For an investment asset with spot price S_0, risk-free rate r, and maturity T:

No income: F_0 = S_0 e^{rT}.
Continuous dividend/income yield q: F_0 = S_0 e^{(r-q)T}. Income earned while holding the asset offsets the financing cost — a stock index paying dividends, or a foreign currency earning its own interest rate q = r_f, need a smaller premium over spot (or even trade at a discount to spot, if q > r).

Nowhere in this argument did anyone forecast where the price is going. The formula depends only on S_0, r, q, and T — quantities observable today. That is the entire point of an arbitrage argument: it pins down a price without anyone needing to be right about the future.

Worked example: pricing a gold forward

Spot gold trades at S_0 = \$1{,}950 per ounce. Six-month dollar interest rates are r = 5\% (continuously compounded), and — treating gold here as a pure investment asset with negligible storage cost — there's no income to net off. The fair six-month forward price is

F_0 = 1{,}950 \times e^{0.05 \times 0.5} = 1{,}950 \times e^{0.025} \approx 1{,}950 \times 1.0253 \approx \$1{,}999.30.

Now add income: a stock index at S_0 = \$4{,}500, one-year risk-free rate r = 4\%, dividend yield q = 2\%:

F_0 = 4{,}500 \times e^{(0.04 - 0.02)\times 1} = 4{,}500 \times e^{0.02} \approx 4{,}500 \times 1.0202 \approx \$4{,}590.90.

The dividend yield shrinks the premium: without it, the same index would carry to 4{,}500 \times e^{0.04} \approx \$4{,}591.90 — a full percentage point of carry, not netted against two percentage points of dividends.

Watching the curve bend

Fix today's spot at S_0 = 100 and plot the fair forward price F_0(T) = 100\,e^{(r-q)T} as maturity T stretches from zero out to two years. With r > q the curve climbs — the classic upward-sloping term structure of forward prices called contango. Push the income yield q above the interest rate r (drag its slider past r's) and the curve tips over and slopes down instead — backwardation. For a currency forward this just means the foreign interest rate exceeds the domestic one; for the storable commodities of the next lesson, a similar downward tilt has a very different — and more interesting — explanation.

The same F_0 = S_0 e^{(r-q)T} skeleton reappears, relabeled, across the whole derivatives world. For a stock index, q is the dividend yield. For a currency forward, covered interest rate parity is exactly this formula with q = r_f, the foreign risk-free rate — holding foreign currency "pays a dividend" equal to the interest it earns abroad. For a storable commodity, as the next lesson shows, the same slot gets filled by a negative yield — a storage cost instead of an income — flipped in sign because now you're paying to carry the asset rather than being paid. Once you've internalized "carry = financing cost minus income," you can read off the pricing formula for almost any forward market by asking only one question: what income or cost does holding the asset actually generate?

It's natural to read F_0 as "the market's best guess of where the price will be at T." That's not what this formula says. F_0 = S_0 e^{(r-q)T} was derived purely from today's observable numbers and a no-arbitrage argument — it says nothing whatsoever about anyone's expectation of the future spot price S_T. If the market's true expected future price were, say, 10% above F_0, the forward price still wouldn't move to match it — because the arbitrage argument only needs riskless borrowing, lending, buying and selling today; it never requires anyone to be right about tomorrow. (There is a separate, deeper relationship between forward prices and expected future spot prices under risk-neutral pricing — but that's a story for Mathematics of Finance, not a contradiction of the arbitrage argument here.)